Citation: Andrea Franceschetti, Andrea Pugliese, Dimitri Breda. Multiple endemic states in age-structured $SIR$ epidemic models[J]. Mathematical Biosciences and Engineering, 2012, 9(3): 577-599. doi: 10.3934/mbe.2012.9.577
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In this paper we focus on the process of angiogenesis, which means formation of new vessels from pre-existing ones. It is a normal and vital process in growth and development of animal organisms, but it is also essential in the transition of avascular forms of solid tumours into metastatic ones. Small tumours (less than 1-2 mm
Mathematical modelling of this process is inextricably linked with the idea of anti-angiogenic treatment first considered by Folkman in 1971 (c.f.[15]). However, as anti-angiogenic agents were not known that time, it took more than 20 years for the specific models of angiogenesis and anti-angiogenic treatment appear (c.f.[17] were one of the best known models of it has been proposed).
Various approaches were used to describe the angiogenesis process in mathematical language. The simplest approach is based on ordinary differential equations describing the dynamics of tumour and vasculature, like in the approach of Hahnfeldt et al.[17]. This idea was extended by many authors, including d'Onofrio and Gandolfi[14], Agur et al.[2], Bodnar and Foryś[8], Poleszczuk et al.[21], Piotrowska and Foryś[20,16]. However, in many cases spatio-temporal dynamics of vessels and tumour structure is important. To reflect such a structure one can use the approach of partial differential equations. First models of that type was based on reaction-diffusion equations with the process of chemotaxis taken into account; c.f.[12]. Many papers focusing on chemotaxis modelling was published by M. Chaplain and coauthors (c.f.[12], [19] where also haptotaxis process was considered, [3] with the influence of external forces). Yet another approach is based on cellular automata. Models of that type was proposed by Rieger and coauthors (c.f.[6,23]), while Alarcón et al.[1] considered hybrid cellular automaton in the context of multiscale modelling. Very nice review of various types of spatio-temporal models of vasculogenesis and angiogenesis processes could be find in[22], where a list of many other references on that topic is available.
In this paper we consider the model of tumour angiogenesis proposed in[7] and studied in[9] in the context of discrete delays. However, discrete delays could be only an approximation of delays present in real life, and therefore, following[10] we decided to incorporate distributed delays and compare the results with those for the discrete case. We mainly focus on the Erlang distributions, however some results for general distributions are also obtained. Partial results for the distributed delay in the vessel formation process was presented in[5]. Here, we incorporate distributed delays both in the vessel formation and tumour growth processes. Thus we consider the following system of the first order differential equations with distributed delays
$
˙N(t)=αN(t)(1−N(t)1+f1(h1(Et))),˙P(t)=f2(E(t))N(t)−δP(t),˙E(t)=(f3(h2(Pt))ds−α(1−N(t)1+f1(h1(Et)))) E(t),
$
|
(1.1) |
where
The functions
$ h_i(\phi) = \int_0^{\infty} k_i(s)\phi(-s){\rm{d}} s, $ |
and we assume
$ \int_0^\infty k_i(s){\rm{d}} s=1 \;\; \text{and}\;\; 0<\int_0^\infty s k_i(s) {\rm{d}} s<\infty, i=1, \, 2. $ |
For any
Moreover, we assume that the functions
(A1)
(A2)
(A3)
For detailed derivation of the model described by Eqs. (1.1) we refer to[7,9].
To close the problem we need to define initial data. Let
$
Φ={ϕ∈C:lims→−∞ϕ(s)es=0andsups∈(−∞,0]|ϕ(s)es|<∞},‖ϕ‖Φ=sups∈(−∞,0]|ϕ(s)es|,
$
|
and we consider initial functions from the set
In this section we consider basic properties of system (1.1) for general distributions
Theorem 2.1. For any arbitrary initial function
$
Nmin≤N(t)≤Nmax,0≤P(t)≤max{A2δNmax,ϕ2(0)},0≤E(t)≤ϕ3(0)exp((B3+α(Nmax−1))t),
$
|
(2.1) |
where
$ N_{\min{}} = \min \left\{1,\phi_N(0)\right\}, N_{\max{}}=\max \left\{\phi_N(0),1+B_1\right\}. $ |
Proof. It is easy to see that the right-hand side of system (1.1) is locally Lipschitz, which yields local existence of the solution of (1.1). Non-negativity follows from the form of this right-hand side. Inequalities (2.1) could be obtained in the same way as in[9]. Then the global existence of the solutions can be proved by the use of Theorem 2.7 from [18,Chapter 2].
Now, we turn to steady states. It is obvious that there are at least two steady states
$ A~= (0, 0, 0) \text{and} B = \Bigl(1, \frac{A_2}{\delta}, 0\Bigr) , $ |
compare [9]. Moreover, there can exist positive steady states
$ g(x)=f_2(x)(1+f_1(x))-\delta m_3 . $ | (2.2) |
Stability of the steady states
Proposition 2.2. The trivial steady state
In this section, we focus on examining the stability of the positive steady states
In the general case let us define
$ K_i(\lambda) = \int_0^\infty k_i(s) {\rm{e}}^{-\lambda s} {\rm{d}} s . $ | (2.3) |
Then, the stability matrix of system (1.1) for the steady state
$
M(\bar{N},\bar{P},\bar{E})=
\left(−α−λ0αd1K1(λ)f2(¯Ei)−δ−λ−¯Ni d2αb¯Eid3¯EiK2(λ)−αbd1¯EiK1(λ)−λ \right),
$
|
where
$ b=\frac{1}{1+f_1(\bar E_i)}, d_1 = f_1'(\bar E_i)>0, d_2 = -f_2'(\bar E_i)>0, d_3 = f_3'(m_3)>0 . $ |
Hence, the characteristic quasi-polynomial has the form
$
W(λ)=λ3+C1λ2+C2λ+(λ2+δλ)C3K1(λ)++(λ+α)C4K2(λ)−C3C5K1(λ)K2(λ),
$
|
(2.4) |
with
$ C_1=\alpha+\delta, C_2=\alpha \delta, C_3=\alpha \beta d_1, C_4=\frac{\beta d_2 d_3}{b^2}, C_5=\delta d_3 m_3, \beta= b\bar{E_i}. $ |
Conditions (A1)-(A3) guarantee positivity of
Theorem 2.3. If
Proof. We show that the characteristic function (2.4) has at least one positive real root. In the proof of Theorem 3.4 in [9] it is shown that the sign of
Now, we focus on the cases when only one of the considered processes is delayed and the other is instantaneous. First, we consider
In the theorem presented below we shall use the following auxiliary polynomials and positive zeros of these polynomials. Let us define
$ w_1(\omega) = -\omega^3-C_3 \omega^2+(C_2+C_4-\delta C_3 )\omega- C_3C_5, $ | (2.5) |
$ w_3(\omega) = -(C_1+C_3)\omega^2-\delta C_3 \omega+\alpha C_4-C_3C_5 , $ | (2.6) |
and
$ w_4(\omega) = -(C_1-C_3)\omega^2+\delta C_3 \omega+\alpha C_4+C_3C_5. $ | (2.7) |
To obtain positive zeros of
$ P_4 = \frac{\delta C_3+ \sqrt{\delta^2C_3^2+4(C_1-C_3)(\alpha C_4+C_3C_2)}}{2(C_1-C_3)}. $ |
In the following, we require
Theorem 2.4. Assume that
(ⅰ)
$ \frac{2 \alpha^2 b}{d_2 \bar{E_i} }<d_3< \min\left \{\frac{b}{2d_2 \bar{E_i}}\left(\delta^2-{\alpha}^{2}\left({\beta}^{2}{d_{{1}}}^{2}-1\right)\right),\ \frac{\delta}{2{\beta}^{2}{d_{{1}}}^{2} m_{{3}}} \left(1-{\beta}^{2}{d_{{1}}}^{2}\right)\right\}, $ |
or
(ⅱ)
then
Proof. The steady state
Let
$ K_1(i \omega)=\eta_1-i\zeta_1, \eta_1=\int\limits_0^\infty k_1(s) \cos(\omega s){\rm{d}} s, \ \ \zeta_1= \int\limits_0^\infty k_1(s) \sin(\omega s){\rm{d}} s. $ |
Thus
$ W(i\omega)=-i\omega^3-C_1\omega^2+i(C_2+C_4)\omega+((-\omega^2+i\delta \omega)C_3 -C_3C_5)(\eta_1-i\zeta_1)+\alpha C_4 , $ |
and we have
$
Re(W(iω))=−C1ω2+αC4−C3(ω2+C5)η1+δC3ωζ1,Im(W(iω))=−ω3+(C2+C4)ω+δC3ωη1+C3(ω2+C5)ζ1.
$
|
(2.8) |
Assume that there exists
$
(−C1ω2+αC4)2+(−ω3+(C2+C4)ω)2=(−C3(ω2+C5)η1+δC3ωζ1)2+(δC3ωη1+C3(ω2+C5)ζ1)2.
$
|
(2.9) |
For Eq. (2.9) we have
$
L.H.S=ω6+(C21−2(C2+C4))ω4+((C2+C4)2−2αC1C4)ω2+α2C24,R.H.S=C23(ω4+(δ2+2C5)ω2+C25)(ζ21+η21).
$
|
Schwarz inequality yields
$
(∞∫0cos(ωs)k1(s)ds)2=(∞∫0cos(ωs)d(∫s0k1(u)du))2≤∞∫0cos2(ωs)d(∫s0k1(u)du)∞∫0d(∫s0k1(u)du)=∞∫0cos2(ωs)k1(s)ds,(∞∫0sin(ωs)k1(s)ds)2≤∞∫0sin2(ωs)k1(s)ds.
$
|
Consequently,
$ 0=L.R.S.-R.H.S. $ |
$
ω6+(C21−2˜C2,4)ω4+((˜C2,4)2−2αC1C4)ω2++α2C24−C23(ω4+(δ2+2C5)ω2+C25),
$
|
where
$
F(y)=y3+(C21−2˜C2,4−C23)y2+((˜C2,4)2−2αC1C4−C23(δ2−2C5))y++α2C24−C23C25,
$
|
Existence of purely imaginary eigenvalue requires
Clearly, the free term is positive due to
$ {\beta}{d_{{1}}}<1, \alpha^2 < {\frac {\beta\,d_{{2}}d_{{3}}}{2{b}^{2}}} \;\;\; \mbox{and }$ |
$ {\delta}>\max \left\{\sqrt{{\alpha}^{2}({\beta}^{2}{d_{{1}}}^{2}-1)+{\frac {2 \beta\,d_{{2}}d_{{3}}}{{b}^{2}}}},\frac{2{\beta}^{2}{d_{{1}}}^{2}d_{{3}}m_{{3}}}{1-{\beta}^{2}{d_{{1}}}^{2} }\right\}. $ |
Therefore,
$ C_1^2-2(C_2+C_4)-C_3^2={\delta}^{2}- {\alpha}^{2} \left( {\beta}^{2}{d_{{1}}}^{2}-1 \right) -{\frac {2 \beta\,d_{{2}}d_{{3}}}{{b}^{2}}} >0 , $ |
$ \;\;\;\;\;(C_2+C_4)^2-2 \alpha C_1 C_4 -\delta^2 C_3^2 -2 C_3^2 C_5= \\ \left( \alpha\,\delta+{\frac {\beta\,d_{{2}}d_{{3}}}{{b}^{2}}} \right) ^{2} -{\frac {2 \alpha\, \left( \delta+\alpha \right) \beta\, d_{{2}}d_{{3}}}{{b}^{2}}} - {\delta}^{2}{\alpha}^{2}{\beta}^{2}{d_{{1}}}^{2}-2{\alpha}^{2}{\beta}^{2}{d_{{1}}}^{2}\delta\,d_{{3}}m_{{3}} = \\\left( \left( 1-{\beta}^{2}{d_{{1}}}^{2} \right) {\alpha}^{2} \delta-2 {\alpha}^{2}{\beta}^{2}{d_{{1}}}^{2}d_{{3}}m_{{3}}\right) \delta+\frac{\beta\,d_{{2}}d_{{3}}}{b^2} \left( {\frac {\beta\,d_{{2}}d_{{3}}}{{b}^{2}}}- 2{\alpha}^{2} \right) >0 . $ |
Due to the continuous dependance of eigenvalues on the model parameters this completes the proof of the first part.
For the proof of the second part, notice that (2.8) implies
$
Re(W(iω))≥−(C1+C3)ω2−δC3ω+αC4−C3C5,Re(W(iω))≤−(C1−C3)ω2+δC3ω+αC4+C3C5,Im(W(iω))≥−ω3−C3ω2+(C2+C4−δC3)ω−C3C5.
$
|
It is clear that,
$
Re(W(iω))>0 for ω∈[0,P3),and Im(W(iω))>0 for ω∈(P1,P2),Re(W(iω))<0 for ω∈(P4,∞).
$
|
Inequalities
Remark 1. If the second condition of assumption (ⅰ) of Theorem 2.4 is satisfied, then
$ \delta^2>3\alpha^2 . $ |
If this inequality holds, we can choose sufficiently small
In the following, we shall consider Erlang distributed delays. The density of Erlang distribution is given by
$
k(s)={am(s−σ)m−1(m−1)!e−a(s−σ),s≥σ,0,otherwise,
$
|
(2.10) |
where
Now, let us consider the first distribution to be Erlang,
$
WI(λ)=(a+λ)m(λ3+C1λ2+(C2+C4)λ+αC4)++am(C3λ2+δC3λ−C3C5)e−λσ.
$
|
(2.11) |
As the case
Theorem 2.5. If
Proof. For this case the characteristic function
$ W_I(\lambda) = (a+\lambda)\Bigl(\lambda^3+C_1\lambda^2 +\bigl(C_2+C_4\bigr)\lambda +\alpha C_4\Bigr) +a\Bigr(C_3\lambda^2+\delta C_3\lambda -C_3C_5\Bigl). $ |
The Routh-Hurwitz Criterion for
$ q_1 q_2 q_3>q_3^2+q_1^2 q_4 , $ | (2.12) |
where
$ q_1 = a+C_1, q_2 = a(C_1+C_3)+\eta_2, q_3 = a(\eta_2+\delta C_3) + \alpha C_4, q_4 = a\eta_4, \ $ |
and
$ \eta_2 = C_2+C_4, \eta_4 = \alpha C_4-C_3C_5 , $ |
Notice that inequality (2.12) is equivalent to
$ P(a) = a_3 a^3 + a_2 a^2 + a_1 a~+ a_0 >0, $ |
where
$
a3=(C1+C3)(η2+δC3)−η4,a2=αC4(C1+C3)+C2(η2+δC3)+C1(C1+C3)(η2+δC3) +C4(η2+δC3)−(η2+δC3)2−2C1η4,a1=C1(η22+αC3C4+δC3η2)−αC4(η2+2δC3)+C21C3C5,a0=αC4(C1η2−αC4).
$
|
Due to the definitions of
$
a0=αC4((δ+α)(C2+C4)−αC4)=αC4((δ+α)C2+δC4)>0,a1=(α+δ)((C2+C4)2+αC3C4+δC3(C2+C4))−αC4(C2+C4)+−2αδC3C4+C21C3C5=α(η2C2+αC3C4+δC2C3)+δη2(η2+δC3)>0,a2=αC4C3+C2(η2+δC3)+(C1(α+δ)+C1C3)(η2+δC3)+C4(η2+δC3)+−(η2+δC3)2−αC1C4+2C3C5,=αC4C3+(δC1+(α+δ)C3)(η2+δC3)+αC1(C2+δC3)+−δC2C3−δC4C3−δ2C23+2C3C5,=αC4C3+(δC1+α)C3(η2+δC3)+αC1(C2+δC3)+2C3C5>0,a3=(δ+α+C3)(C2+C4+δC3)−αC4+C3C5=(δ+C3)(C2+C4+δC3)+α(C2+δC3)+C3C5>0.
$
|
Hence,
Remark 2. Although Theorem 2.4 gives condition guaranteeing stability of the positive steady state for any delay distribution, Theorem 2.5 says that the positive steady state, if it is stable for the case without delay, cannot lose stability when the tumour growth process is delayed according to the Erlang distribution.
Now, we switch to the case when
$
WII(λ)=(a+λ)m(λ3+(C1+C3)λ2+(C2+δC3)λ)++am(C4λ+αC4−C3C5)e−λσ.
$
|
(2.13) |
Again, because the case
Proposition 2.6. If
$
Q1=C1+C3+a,Q2=C2+δC3+a(C1+C3),Q3=a(C2+C4+δC3),Q4=a(αC4−C3C5),
$
|
then the positive steady state
Proof. For
$
WII(λ)=λ4+(C1+C3+a)λ3+(C2+δC3+a(C1+C3))λ2++a(C2+C4+δC3)λ+a(αC4−C3C5),
$
|
and the assertion of the proposition comes directly from the Routh-Hurwitz Criterion.
Now, we try to answer the question when the assumptions of Proposition 2.6 are satisfied. To simplify calculations and shorten notation, let us denote
$ \eta_1 = C_1+C_3, \eta_2 = C_2+\delta C_3, \eta_4 = \alpha C_4-C_3C_5. $ |
With this notation we have
$ Q_1=\eta_1+a, Q_2=\eta_2+a \eta_1, Q_3=a(\eta_2+C_4), Q_4 =a \eta_4,$ |
$ Q_i>0 \ \ \text{for} \ \ i=1,\ldots ,4. $ |
Now, the Routh-Hurwitz condition is equivalent to
$
a2(η1(η2+C4)−η4)+a((η2+C4)(η21−C4)−2η1η4)++η1(η2(η2+C4)−η1η4)>0.
$
|
(2.14) |
Notice that the coefficient of
$ \eta_1 \bigl(\eta_2+C_4\bigr)-\eta_4 = \eta_1\eta_2 + (\alpha+\delta+C_3)C_4 - \alpha C_4+C_3C_5 =\eta_1\eta_2+(\delta+C_3)C_4+C_3C_5>0. $ |
Now, we have only three possibilities:
1.
2.
3.
To obtain two changes of stability, we need to have
$ \Bigl(\bigl(\eta_2+C_4\bigr)\bigl(\eta_1^2-C_4\bigr)-2\eta_1\eta_4\Bigr)^2>4 \eta_1\Bigl(\eta_2\bigl(\eta_2+C_4\bigr)-\eta_1\eta_4\Bigr)\Bigl(\eta_1 \bigl(\eta_2+C_4\bigr)-\eta_4\Bigr) , $ | (2.15) |
together with
$ \frac{(\eta_1^2-C_4)(\eta_2+C_4)}{2}<\eta_1\eta_4< {\eta_2(\eta_2+C_4)}. $ |
Inequality (2.15) is equivalent to
$ \bigl(\eta_2+C_4\bigr)\bigl(\eta_1^2-C_4\bigr)^2 -4\eta_1\eta_4\bigl(\eta_1^2-C_4\bigr) -4 \eta_1^2\eta_2\bigl(\eta_2+C_4\bigr)+4\eta_1\eta_2\eta_4 +4\eta_1^3\eta_4 >0 , $ |
and collecting terms with
$ \bigl(\eta_2+C_4\bigr)\bigl(\eta_1^2-C_4\bigr)^2 -4\eta_1\eta_4\bigl(\eta_1^2-C_4\bigr) +4\eta_1 \Big( \eta_4 \bigl(\eta_2 +\eta_1^2\bigr) - \eta_1\eta_2\bigl(\eta_2+C_4\bigr)\Big) >0 . $ | (2.16) |
Notice that the free and linear terms of (2.16) are positive under the assumption
$ \eta_4> \frac{ \eta_1\eta_2\bigl(\eta_2+C_4\bigr)}{\eta_2+\eta_1^2} \text{and} \eta_1^2<C_4 . $ |
We have
Eventually, two stability switches are possible under the assumptions
$ \frac{ \eta_1\eta_2\bigl(\eta_2+C_4\bigr)}{\eta_2+\eta_1^2} <\eta_4<\frac{\eta_2(\eta_2+C_4)}{\eta_1} \text{and} \eta_1^2<C_4 . $ |
Proposition 2.7. If
(ⅰ)
(ⅱ)
Proof. For the characteristic function
$
F(y)=y(a2+y)m(y2+((C1+C3)2−2(C3δ+C2))y+(C3δ+C2)2)−−a2mC24y−a2m(C4α−C3C5)2,
$
|
where
$ (C_1+C_3)^2-2(C_3\delta+C_2)=\alpha^2(1+\beta d_1)^2+\delta^2>0. $ |
Clearly,
Eventually, we consider both distributions to be Erlang with the same parameters.
Proposition 2.8. If
$ (q_{11}q_{44}-q_{55})(q_{11}q_{22}q_{33}-q_{33}^2-q_{11}^2q_{44})>q_{55}(q_{11}q_{22}-q_{33})^2+q_{11}q_{55}^2, $ |
where
$
q11=C1+2a ,q22=C2+2a C1+a2+aC3,q33=2a C2+a2C1+a2C3+aC3δ+aC4,q44=a2C2+a2C3δ+a2C4+aC4α,q55=a2C4α−a2C3C5,
$
|
then the positive steady state
Proof. For
$
W(λ)=1(a+λ)2((λ3+C1λ2+C2λ)(a+λ)2+a(λ2+δλ)C3(a+λ)++a(λ+α)C4(a+λ)−a2C3C5).
$
|
Hence, we need to study roots of the polynomial
$ \lambda^5+(C_1+2 a~)\lambda^4+(C_2+2 a~C_1+a^2+a C_3)\lambda^3+(2 a~C_2+a^2 C_1+a^2 C_3+a C_3 \delta+a C_4)\lambda^2+\\ +(a^2 C_2+a^2 C_3 \delta +a^2 C_4+a C_4 \alpha) \lambda+a^2 C_4 \alpha-a^2C_3C_5=0. $ |
A direct application of the Routh-Hurwitz Criterion completes the proof.
For the numerical simulations we choose functions
$f1(E)=b1Enc1+En,f2(E)=a2c2c2+E,f3(P)=b3(P2−m23)m23b3a3+P2, $
|
and
$ a_2 = 0.4, \ \ a_3=1, \ \ b_1 = 2.3, \ \ b_3=1, \ \ c_1 = 1.5, \ \ c_2 = 1, \ \ \alpha = 1, \ \ \delta = 0.34. $ | (3.1) |
For these values of parameters there exist three positive steady states:
$ D_1\approx(1.04, 1.05, 0.17), D_2\approx (1.37, 1.05, 0.54), D_3\approx (2.67, 1.05, 1.99) . $ |
Now, we can influence the model dynamics changing the value of
In Table 1 we presented critical values of delay at which the steady states
steady state |
steady state |
||||||||||
0.332 | 0.346 | 0.36 | 0.368 | 0.378 | 0.3 | 0.332 | 0.346 | 0.36 | 0.368 | ||
discrete | 66.7 | 33.4 | 29.3 | 43.6 | 182 | 4.49 | 5.89 | 7.53 | 13.0 | 94.0 | |
steady state does not lose stability | |||||||||||
176 | 54.7 | 69.1 | 106.1 | 460 | 5.58 | 9.36 | 14.4 | 32.2 | 284 | ||
89.9 | 29.6 | 37.4 | 56.6 | 234 | 4.03 | 5.97 | 8.34 | 16.6 | 135 |
In Fig. 2 we see exemplary solutions of system (1.1) for parameters given by (3.1) and
For comparison, we present solutions of system (1.1) with Erlang distributed delay with parameters
From our numerical analysis it is clear that the most robust is the model with Erlang distribution with
In this paper a model of tumour angiogenesis with distributed delays was considered. We proved basic mathematical properties of the model showing that solutions are unique, non-negative and well defined on the whole positive half-line. We formulated conditions on the model parameters that guarantee lack of change of local stability of a steady state for any distribution of delays. On the other hand, we proved condition under which stability change can take place and Hopf bifurcation occurs. We gave more strict conditions in the case when delays are distributed according to Erlang distributions. Our results indicate that the model with distributed delays is more stable than with discrete ones. In particular, we observe stabilisation of the solution in a steady state value for some delay distributions while in the same time solutions of the model with discrete delays exhibit oscillations. In the case of Erlang distributions we observe that the behaviour of the solution for large shape parameter is closer to the behaviour of the solution to the model with discrete delays.
The model considered in this paper is an extension of the model proposed earlier by Agur et. al.[2]. In this paper Agur et al. tried to simplified more complex computer model of angiogenesis process proposed in[4]. However, this model always exhibits oscillatory dynamics, which is not realistic. On the other hand, according to the data presented in[4], such type of the dynamics should be also present in the model, and therefore they included time delays into their model. Our idea was to combine the properties of the Hahnfeldt et al. model with the properties of the one presented in[2]. Here we decided to use distributed delays instead of discrete ones in order to make the model more realistic comparing to the previous discrete case[7]. Our results show that both type of the model dynamics could be observed for the model with delays distributed according to Erlang distributions, depending on the shape parameter, which is good from the point of view of potential applications. Although we have not validated our model with experimental data, it is done for a small modification of this model and the results should be published shortly; c.f.[11].
1. | Krzysztof Psiuk-Maksymowicz, 2024, Chapter 17, 978-3-031-38429-5, 215, 10.1007/978-3-031-38430-1_17 |
steady state |
steady state |
||||||||||
0.332 | 0.346 | 0.36 | 0.368 | 0.378 | 0.3 | 0.332 | 0.346 | 0.36 | 0.368 | ||
discrete | 66.7 | 33.4 | 29.3 | 43.6 | 182 | 4.49 | 5.89 | 7.53 | 13.0 | 94.0 | |
steady state does not lose stability | |||||||||||
176 | 54.7 | 69.1 | 106.1 | 460 | 5.58 | 9.36 | 14.4 | 32.2 | 284 | ||
89.9 | 29.6 | 37.4 | 56.6 | 234 | 4.03 | 5.97 | 8.34 | 16.6 | 135 |
steady state |
steady state |
||||||||||
0.332 | 0.346 | 0.36 | 0.368 | 0.378 | 0.3 | 0.332 | 0.346 | 0.36 | 0.368 | ||
discrete | 66.7 | 33.4 | 29.3 | 43.6 | 182 | 4.49 | 5.89 | 7.53 | 13.0 | 94.0 | |
steady state does not lose stability | |||||||||||
176 | 54.7 | 69.1 | 106.1 | 460 | 5.58 | 9.36 | 14.4 | 32.2 | 284 | ||
89.9 | 29.6 | 37.4 | 56.6 | 234 | 4.03 | 5.97 | 8.34 | 16.6 | 135 |